### Abstract

We analyze a class of dynamical systems of the type ȧ_{n}(t) = c_{n-1} a_{n-1}(t) - c_{n} a_{n+1}(t) + f _{n}(t), n ∈ ℕ, a 0=0, where f _{n} (t) is a forcing term with f_{n}(t) ≠ = 0 only for ≤n n_{*} < ∞ and the coupling coefficients c _{n} satisfy a condition ensuring the formal conservation of energy 1/2 Σ_{n} |a _{n}(t)|^{2}. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f _{n} (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c _{n} . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E |a_{n}|^{2} scales as n^{-α} as n→∞. Here the exponents α depend on the coupling coefficients c _{n} and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.

Original language | English (US) |
---|---|

Pages (from-to) | 189-220 |

Number of pages | 32 |

Journal | Communications in Mathematical Physics |

Volume | 276 |

Issue number | 1 |

DOIs | |

State | Published - Nov 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*276*(1), 189-220. https://doi.org/10.1007/s00220-007-0333-0

**Simple systems with anomalous dissipation and energy cascade.** / Mattingly, Jonathan C.; Suidan, Toufic; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 276, no. 1, pp. 189-220. https://doi.org/10.1007/s00220-007-0333-0

}

TY - JOUR

T1 - Simple systems with anomalous dissipation and energy cascade

AU - Mattingly, Jonathan C.

AU - Suidan, Toufic

AU - Vanden Eijnden, Eric

PY - 2007/11

Y1 - 2007/11

N2 - We analyze a class of dynamical systems of the type ȧn(t) = cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n n* < ∞ and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy 1/2 Σn |a n(t)|2. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E |an|2 scales as n-α as n→∞. Here the exponents α depend on the coupling coefficients c n and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.

AB - We analyze a class of dynamical systems of the type ȧn(t) = cn-1 an-1(t) - cn an+1(t) + f n(t), n ∈ ℕ, a 0=0, where f n (t) is a forcing term with fn(t) ≠ = 0 only for ≤n n* < ∞ and the coupling coefficients c n satisfy a condition ensuring the formal conservation of energy 1/2 Σn |a n(t)|2. Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely E |an|2 scales as n-α as n→∞. Here the exponents α depend on the coupling coefficients c n and E denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.

UR - http://www.scopus.com/inward/record.url?scp=34848825911&partnerID=8YFLogxK

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U2 - 10.1007/s00220-007-0333-0

DO - 10.1007/s00220-007-0333-0

M3 - Article

VL - 276

SP - 189

EP - 220

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -